Change or update audio start with video texture start frame
Here is a workaround solution

A custom property, named "audio" is added to the movie texture, with a value of the soundstrip name.  Every time the script is run it updates the audio strip, pointed to by the audio key, to have the same frame_start as the tex.

import bpy

context = bpy.context
scene = context.scene
sequences = scene.sequence_editor.sequences_all
# all the textures with audio custom prop
textures = [t for t in bpy.data.textures if "audio" in t.keys()]

for t in textures:
    audio_strip = sequences.get(t["audio"])
    if audio_strip:
        audio_strip.frame_start = t.image_user.frame_start